1. ## Question about linear combination of two vectors

Question:
"What condition must be met by the two vectors of a basis such that any given vector can be expressed as a linear combination of these two vectors?"

"They must be linearly independent (or non-collinear)."
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Before checking the answer, I answered the exact opposite. My argument is that, when you combine two vectors in order to form another with a linear combination, those two vectors had to be collinear in order for them to produce another vector in the same linear path (=same slope).

How am I wrong?

2. Originally Posted by s3a
Question:
"What condition must be met by the two vectors of a basis such that any given vector can be expressed as a linear combination of these two vectors?"

"They must be linearly independent (or non-collinear)."
----------------------------------------------------------------------------------------------------------------------------------

Before checking the answer, I answered the exact opposite. My argument is that, when you combine two vectors in order to form another with a linear combination, those two vectors had to be collinear in order for them to produce another vector in the same linear path (=same slope).

How am I wrong?
The general concept is: If you have two non-collinear vectors $\displaystyle \vec u$ and $\displaystyle \vec v$ all other vectors in the same plane can be expressed as:

$\displaystyle \vec r = a \cdot \vec v + b \cdot \vec u$

where the coefficients a and b are of degree 1 that means linear.

In other words: The expression linear is refering to the degree of the coefficients.

3. Originally Posted by s3a
Question:
"What condition must be met by the two vectors of a basis such that any given vector can be expressed as a linear combination of these two vectors?"